Incidence conjecture for Klein's arrangements of lines and conics

Let $\mathrm{GR}(21_4)$ be the Grünbaum–Rigby configuration, decomposed into subconfigurations $\mathrm S_{12}$, $\mathrm S_{13}$, and $\mathrm S_{23}$ with point orbits $\mathrm P_1$, $\mathrm P_2$, and $\mathrm P_3$. Let $\mathrm S'_{12}$, $\mathrm S”_{13}$, and $\mathrm S”'_{23}$ be configurations isomorphic to these subconfigurations, with point sets $\mathrm P'_1,\mathrm P'_2\subset\mathrm S'_{12}$, $\mathrm P”_1,\mathrm P”_3\subset\mathrm S”_{13}$, and $\mathrm P”'_2,\mathrm P”'_3\subset\mathrm S”'_{23}$. Suppose that the point sets $\mathrm P'_1$ and $\mathrm P'_2$ are inscribed in conics $C_1$ and $C_2$, respectively. Incidence conjecture. The configurations $\mathrm S'_{12}$, $\mathrm S”_{13}$, and $\mathrm S”'_{23}$ can be chosen so that their union forms a configuration isomorphic to $\mathrm{GR}(21_4)$, with $\mathrm P'_1$ identified with $\mathrm P”_1$, $\mathrm P'_2$ identified with $\mathrm P”'_2$, and $\mathrm P”_3$ identified with $\mathrm P”'_3$; moreover, the common point set $\mathrm P”_3=\mathrm P”'_3$ is inscribed in a conic $C_3$. This conjecture asserts that two prescribed conic incidences among the point orbits can be completed compatibly to the full Grünbaum–Rigby configuration, while the existence and geometric properties of the third conic remain the substantive issue.